Causal Structural Realism in Canonical Quantum Gravity

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Causal Structural Realism in Canonical Quantum Gravity

1

A dilemma for the emergence of spacetime in canonical quantum gravity

Vincent Lam[*][†] & Michael Esfeld[‡]

Abstract

The procedures of canonical quantization of the gravitational field apparently lead to entities for which any interpretation in terms of spatio-temporal localization or spatio-temporal extension seems difficult. This fact is the main ground for the suggestion that can often be found in the physics literature on canonical quantum gravity according to which spacetime may not be fundamental in some sense. This paper aims to investigate this radical suggestion from an ontologically serious point of view in the cases of two standard forms of canonical quantum gravity, quantum geometrodynamics and loop quantum gravity. We start by discussing the physical features of the quantum wave functional of quantum geometrodynamics and of the spin networks (and spin foams) of loop quantum gravity that motivate the view according to which spacetime is not fundamental. We then point out that, by contrast, for any known ontologically serious understanding of quantum entanglement, the commitment to spacetime seems indispensable. Against this background, we then critically discuss the idea that spacetime may emerge from more fundamental entities. As a consequence, we finally suggest that the emergence of classical spacetime in canonical quantum gravity faces a dilemma: either spacetime ontologically emerges from more fundamental non-spatio-temporal entities or it already belongs to the fundamental quantum gravitational level and the emergence of the classical picture is merely a matter of levels of description. On the first horn of the dilemma, it is unclear how to make sense of concrete physical entities that are not in spacetime and of the notion of ontological emergence that is involved. The second horn runs into the difficulties raised by the physics of canonical quantum gravity.

Keywords: spacetime, emergence, quantum geometrodynamics, loop quantum gravity, quantum superposition, quantum entanglement, non-separability, structural realism, local beables.

1.Introduction

The ontological status of space and time (or spacetime) and their relationship with matter are among the most important and crucial issues in philosophy of nature as well as in fundamental physics. One of the central aims of a theory of quantum gravity (QG) is to describe in a consistent way the relationship (‘interaction’) between matter as conceived by quantum field theory (QFT) – that is, quantum fields – and dynamical spacetime as treated by general relativity (GR). As a consequence, a QG theory is expected to provide a better understanding of the fundamental nature of spacetime.

This last point is especially salient in the framework of canonical QG, where canonical quantization is applied to GR, that is, to the gravitational (or metric) field. Since this latter encodes the spacetime geometry in the classical theory, its quantized version can be expected to reveal more fundamental (quantum) aspects of spacetime and gravitation. However, the procedures of canonical quantization of the gravitational field apparently lead to entities for which any interpretation in terms of spatio-temporal localization or spatio-temporal extension seems difficult. This fact is the main ground for the suggestion that can often be found in the physics literature on canonical QG according to which spacetime may not be fundamental in some sense (see for instance the standard textbooks on canonical QG, Rovelli 2004, ch. 10, and Kiefer 2007, ch. 5). This paper aims to investigate this radical suggestion from an ontologically serious point of view in the ‘concrete’ cases of two standard forms of canonical QG, quantum geometrodynamics and loop quantum gravity (LQG), with applications to cosmology.

We start by discussing the physical features of the quantum wave functional of quantum geometrodynamics and of the spin networks (and spin foams) of LQG that motivate the view according to which spacetime is not fundamental (section 2). We then point out that, by contrast, for any known ontologically serious understanding of quantum entanglement, the commitment to spacetime seems indispensable (section 3). Against this background, we then critically discuss the idea that spacetime may emerge from more fundamental entities (section 4). As a consequence, we finally suggest that the emergence of classical spacetime in canonical QG faces a dilemma (section 5): either spacetime ontologically emerges from more fundamental non-spatio-temporal entities or it already belongs to the fundamental QG level and the emergence of the classical picture is merely a matter of levels of description. On the first horn of the dilemma, it is unclear how to make sense of concrete physical entities that are not in spacetime and of the notion of ontological emergence that is involved. The second horn runs into the difficulties raised by the physics of QG.

To the extent that they rely on on-going research programmes rather than well-defined (and well-tested) physical theories, our investigations in this paper remain speculative; however, we argue that they constitute mandatory attempts at providing an interpretative framework that is metaphysically serious about one of the major conceptual challenges raised by quantum gravity, namely the ontological status of spacetime at the fundamental level.

2.The disappearance of spacetime in canonical quantum gravity

2.1Quantum geometrodynamics

Quantum geometrodynamics is the most straightforward canonical quantization of GR (cast in the constrained Hamiltonian formalism) in the sense that it naturally considers the 3-metric and its conjugate momentum (functional of the corresponding extrinsic curvature) as the canonical variables. The quantum Hamiltonian constraint acts on (wave) functionals of the 3-metric (and of the matter fields), giving rise to the famous Wheeler-DeWitt equation. Within this framework, the claim that spacetime is not fundamental is mainly based on four interrelated aspects of quantum geometrodynamics.

  • (QGeom-1) The Wheeler-DeWitt equation does not contain any explicit time parameter, unlike the Schrödinger equation for instance; accordingly – and putting aside the mathematical difficulties related to the Wheeler-DeWitt equation – it is difficult to interpret this equation in dynamical terms, as describing the temporal evolution of a physical system (such as a 3-metric). Of course, that’s one aspect of the ‘problem of time’, which finds its roots in the background independence and diffeomorphism invariance of classical GR already. One way to deal with this ‘problem of time’ is to accept that it indicates that time might not be a fundamental feature of the world. An analogous argument (with respect to the momentum or 3-diffeomorphism constraint) can be brought up about space as well (although without any analogous ‘problem of space’).
  • (QGeom-2) The wave functional of quantum geometrodynamics is not defined on spacetime, but on the (configuration) space of all 3-metrics (if matter is considered, it might also depend on the matter degrees of freedom). It is not at all clear how to relate a wave functional on configuration space satisfying the Hamiltonian and momentum constraints to any description of (classical) spacetime. Indeed the obstacles are many. The wave functional does not determine any curve in configuration space that would straightforwardly correspond to some spacetime. The quantum wave functional is typically understood in terms of probabilities, whose meaning is conceptually challenging in this context (the measurement problem is especially acute in this context), making the link with a spacetime picture rather elusive (Butterfield and Isham 1999, § 5.3). Moreover, on the technical side, the difficulties to construct any mathematically well-defined inner-product for the wave functionals infect the very definition of quantum-mechanical probability (Kiefer 2007, § 5.2.2).
  • (QGeom-3) The non-vanishing commutator between the canonical variables, that is, between the 3-metric and the corresponding functional of the extrinsic curvature, which describes the embedding of the 3-metric into spacetime, seems to forbid any (classical) spacetime understanding (analogous to the fate of quantum particle trajectories in quantum mechanics; see Kiefer 2007, § 5.1).
  • (QGeom-4) Strictly speaking, the full wave functional of the universe (i.e. including all gravitational and matter degrees of freedom) is a huge quantum superposition of many components (which may possibly each receive a ‘quasi-classical’ understanding under certain conditions, see below). It seems difficult to see in what sense such superposition of (functionals of) 3-metrics can be said to describe space(time).

These aspects highlight two types of difficulties for any spatio-temporal understanding: first, the aspects related to the background independence and the dynamics of the theory (QGeom-1) and second, the aspects related to the interpretative issues in quantum theory (QGeom-2, -3, -4) (these latter being aggravated by the former, see below). Let us now turn to the other standard form of canonical quantum gravity.

2.2Loop quantum gravity

One of the main motivations for considering alternative canonical variables for the canonical quantization of GR is to alleviate the mathematical difficulties that plague the constraints of quantum geometrodynamics – in particular the non-polynomial dependence of the constraints on the 3-metric. The choice of the ‘Ashtekar variables’ as canonical variables – basically a SU(2) connection and its corresponding densitized tetrad – lies at the basis of the LQG programme: indeed it opens new and mathematically more rigorous perspectives on the quantum constraints expressed in terms of these variables. The central variables of LQG are the spin network states (suitably defined ‘cylindrical’ (wave) functionals of the SU(2) connection associated with abstract graphs carrying irreducible representations of SU(2)), which form an orthonormal basis in the kinematical Hilbert space of the theory. This latter space can be constructed so that its states are invariant under ‘local’ SU(2) gauge transformations and under 3-diffeomorphisms, i.e. so that the Gauss and (LQG version of the) momentum quantum constraints are satisfied; this kinematical Hilbert space is interpreted as the space of the quantum states of the 3-gravitational field (matter degrees of freedom can be in principle incorporated in the description – very roughly, attaching irreducible representations of the relevant gauge groups to the links of the spin networks). At the kinematical level, several important aspects of LQG ground the claim that space is not fundamental (although such explicit claim about space at the kinematical level is specific to LQG, these aspects – apart from (LQG-1) below – are basically LQG counterparts of those discussed within quantum geometrodynamics). We highlight two of them, which are important for the discussion.

  • (LQG-1) One of the most important results of LQG is that area and volume operators can be defined on the kinematical Hilbert space and turn out to have discrete spectra (spin network states being eigenstates of these geometrical operators). This result is naturally interpreted as an indication of the fundamental discrete nature of space, which (to some extent) can be pictorially represented by spin network graphs, with the nodes representing ‘quanta’ (or ‘atoms’) of space (3-volume) and links representing ‘quanta’ of surfaces (2-surface) separating the ‘quanta’ of volume attached to the corresponding nodes. This discreteness often lies at the heart of the claim about the non-fundamentality of space within LQG (e.g. Wüthrich 2011). What is meant with this claim is that the smooth structure of space as described within classical GR is not fundamental (but may be only approximate in some sense). But this claim alone does not imply that space itself is (ontologically) non-fundamental. The discrete spectra of the geometrical operators are an important (and specific) consequence – prediction – of the theory (rather than a mere assumption as in lattice QFT), and it might be indeed tempting to consider the discrete spin network picture of LQG as evidence that the theory takes space to be a fundamental entity, albeit a discrete one. However, as Rovelli (2011, § 2.4) points out, one should be careful with any too literal reading of such a geometrical picture within this context (there are different – not necessarily ontologically equivalent – geometrical understandings of the spin networks). But the main objection against the fundamentality of space (and spacetime) comes from a more basic quantum feature of the theory.
  • (LQG-2) A generic quantum state at the kinematical level of the theory (i.e. a quantum state of ‘space’ or of the 3-gravitational field) is not a spin network state, but a quantum superposition thereof. As a consequence, the just mentioned geometrical (and intuitive) interpretation of spin networks (literally) describing discrete space breaks down. It is not at all clear in what sense such quantum superpositions could still be considered as representing physical, 3-dimensional space (Rovelli 2004, § 6.7.1 and § 10.1.3 makes the point clear).

This last aspect is of central importance: the basic quantum feature of linear superposition can be convincingly argued to constitute one of the major obstacles for considering space(time) as fundamental within canonical quantum gravity. The crucial point is common to LQG and quantum geometrodynamics: (LQG-2) is the LQG and kinematical version of (QGeom-4), and the argument from quantum superposition can also be raised against spacetime at the dynamical level of LQG.

The discussion about spacetime necessarily involves the dynamical part of the theory, in particular the LQG version of the quantum Hamiltonian constraint and of the Wheeler-DeWitt equation. How to solve this constraint (equation) remains an open issue, but the situation is arguably mathematically more rigorous than in the quantum geometrodynamical case (e.g. a Hamiltonian or Wheeler-DeWitt constraint operator can be rigorously constructed to some extent, see Thiemann 2007, ch. 10). As can be expected, the implementation of this constraint operator displays no explicit time parameter and cannot be easily interpreted in terms of temporal evolution. Indeed, the situation is much analogous to the case of quantum geometrodynamics, see (QGeom-1) above.

Although this paper is mainly concerned with the canonical approach to quantum gravity, we would like to mention the fact that the issues about the non-fundamentality of time (and space) within LQG can be considered from a covariant point of view as well. (The links and convergences between the canonical and covariant frameworks of LQG in recent years have prompted Rovelli 2011 to argue that LQG should better be considered as a theory on its own rather than ‘only’ a quantized version of GR.) Indeed, the dynamics of the theory can be fully understood in terms of transition probability amplitudes for spin network states. The transition amplitudes do not depend explicitly on time and cannot be understood as transition amplitudes between states in time (as in standard, background dependent QFT), even if they can be understood as transition amplitudes between spin network states – from a covariant point of view, a single spin network state, understood as a boundary state (think of a finite spacetime region bounded by some 3-space), can be directly associated with such an amplitude. Within this framework, it is useful to consider the ‘dynamics’ of spin networks and the relevant transition probability amplitudes in terms of spin foams, which can be understood intuitively as higher-dimensional (Feynman-type) ‘graphs’ (‘2-complexes’) describing the ‘evolution’ of the spin networks (‘1-complexes’). The transition amplitude between two spin network states (or for a single ‘boundary’ spin network) is then understood in terms of a sum over the amplitudes of the spin foams with the corresponding boundary. The difficulty to relate the spin foam account of the spin network ‘dynamics’ to any description of spacetime can be highlighted by the analogy with Feynman’s path integrals in QM: the (possibly discrete) spacetimes associated with the spin foams have much the same status as the paths (trajectories) in the path integral version of QM. Note that this is the covariant version of the difficulties with time (and space) already encountered above. We close this section by further noting that the LQG counterparts of (QGeom-2) and (QGeom-3) can also be explicitly raised against the fundamentality of spacetime within this theory.

3.The indispensability of spacetime in the worked out ontologies of quantum physics

3.1Non-separability, entanglement, and structural realism

One of the main arguments for the disappearance of spacetime in canonical QG draws on the superposition principle (see QGeom-4 and LQG-2 above). Whatever the ontological significance of the superposition principle may be, it is the basis for quantum entanglement (which seems indeed prevalent at the QG level, see section 4.1). There is ample evidence from experiments on entangled quantum systems that rules out any ignorance interpretation of quantum theory in terms of an underlying ontology of classical particles. Even in Bohm’s theory, which matches the predictions of textbook quantum mechanics on the basis of an ontology of particles with definite trajectories in spacetime, entanglement is taken into account in terms of a non-classical, holistic connection among the particles, although the superposition principle does not have any ontological significance in Bohm’s quantum theory (see below for a brief discussion of the main interpretations). Schrödinger famously called entanglement with good reason not “one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought” (1935, p. 555).

What does entanglement mean for a concrete ontology for quantum systems? The common philosophical understanding of entanglement invokes a principle of non-separability. The notion of non-separability provides for a concrete physical ontology of entanglement in terms of the following two conditions: (a) two or more quantum systems are non-separable iff, despite their being separated in space, it is not possible to attribute a state to each of them that completely specifies its dynamical properties (see Howard 1985 and Healey 1989 for specifications of the principle of non-separability). (b) Why is this not possible? Because the development of the dynamical properties of the two or more systems is tied together. The evidence for this is that the manipulation of one system changes the probabilities for measurement outcomes on the other systems, as shown by the EPR-Bohm experiments on pairs of quantum systems of spin 1/2 (or pairs of photons) in the singlet state. The theorem of John Bell (1964, reprinted in Bell 1987, ch. 2) proofs that it is not possible to account for the non-local correlations manifested in these experiments in terms of the preparation of the singlet state at the source of the experiment being the common cause of these correlations.

Hence, the ontology of quantum entanglement that has been well developed in the literature on non-relativistic quantum mechanics provides for a clear and serious ontological view under the following two presuppositions: